Archive for the ‘Number Theory’ Category
The Probability That Two Numbers Are Coprime.
tuesday, july 10th, 2012In this post, we derive an explicit lower bound for the number of pairs of coprime integers bounded by some n.
Solving Certain Linear Systems over the Integers.
friday, june 17th, 2011We present a (well-known) method to compute a solution to the linear system Ax=b over the integers, when it is known that the determinant of A is non-zero and that a solution with integral coefficients exists. We also provide a running time analysis.
A Strange Inequality.
thursday, december 9th, 2010We show how to prove a number theoretic inequality, originating from geometry, using an elementary approach.
Rigorous Arithmetic in the Arakelov Divisor Class Group of a Number Field.
tuesday, july 27th, 2010This post presents a poster of mine presented at the poster session of the 9th Algorithmic Number Theory Symphoisum.
How to Compute the 5-adic Expansion of 1/2; or: Hensel’s Lemma and (Non-Analytic) Newton Iteration.
saturday, february 6th, 2010In this post, we consider the quest of computing the 5-adic expansion of 1/2. We begin with introducing p-adic integers and numbers, and discussing when certain polynomials with coefficients in the integers have zeroes in the p-adic integers. This question is closely related to Hensel’s lemma, which can be proven using an algebraic version of Newton’s iteration. We use this to compute approximations of rational numbers in the p-adics, and consider which p-adic numers have an eventually periodic expansion.
Finding Lattice Points, Finite Abelian Groups, and Explaining Algorithms.
friday, january 29th, 2010We compare the tasks of finding points of a lattice, computing the structure of finite abelian groups and explaining algorithms. We show up relations between these three topics and, as an example, depict the baby-step giant-step algorithm for order computation, as well as Terr’s modification of this algorithm.
Obtaining Infrastructures from Global Fields.
tuesday, july 21st, 2009We show how to obtain n-dimensional infrastructures from global fields of unit rank n. We will also discuss how to obtain baby steps in these cases, and show graphical representations of certain two-dimensional infrastructures obtained from function fields.
How to Obtain Reduction Maps for n-dimensional Infrastructures.
tuesday, july 21st, 2009We explain a general technique to obtain a reduction map, given X and d and, varying with the method of construction, additional information for every x in X. Moreover, we explain a technique on how to obtain n-dimensional infrastructures from certain lattices in (n+1)-dimensional space.
n-dimensional Infrastructures.
monday, july 20th, 2009We will introduce n-dimensional infrastructures and briefly discuss reductions, f-representations and giant steps. We will also discuss how infrastructures can be obtained from finite abelian groups.
Interpreting One-dimensional Infrastructures as Groups: f-Representations.
monday, july 20th, 2009We introduce the notion of f-representations and relate them to reduction maps. Moreover, we equip a set of f-representations with a group operation which can be computed purely with baby steps, giant steps and relative distances.
